Problem 28 - Algebra 2 and Trigonometry

Question

Sketch one cycle of each function. \(y=\cos x\)

Step-by-Step Solution

Answer

The cosine function \( y = \cos x \) has one cycle from \( x = 0 \) to \( x = 2\pi \).

Step 1: Understanding the Function

The function given is \( y = \cos x \). It is a trigonometric function that describes the cosine wave. This function has a wave-like shape that repeats every \( 2\pi \) radians.

Step 2: Determine the Period

The period of the cosine function \( y = \cos x \) is \( 2\pi \). This means the function completes one full cycle as \( x \) goes from \( 0 \) to \( 2\pi \).

Step 3: Identify Key Points

For \( y = \cos x \), the key points in one cycle occur at: \( x = 0 \), where \( y = 1 \); \( x = \pi/2 \), where \( y = 0 \); \( x = \pi \), where \( y = -1 \); \( x = 3\pi/2 \), where \( y = 0 \); and \( x = 2\pi \), where \( y = 1 \).

Step 4: Plot the Key Points

Plot the points determined in Step 3 on a coordinate plane: start at \( (0,1) \), go through \( (\pi/2, 0) \), \( (\pi, -1) \), \( (3\pi/2, 0) \), and end at \( (2\pi, 1) \). These points mark the maximum, minimum, and zeros of the function within one cycle.

Step 5: Draw the Curve

Connect the plotted key points with a smooth, continuous curve to form the characteristic wave of the cosine function. This curve will begin at the maximum value, descend through zero, reach the minimum, and rise back to the starting maximum.

Key Concepts

Cosine Wave

The cosine wave is a fascinating and repeatable pattern generated by the trigonometric function \(y = \cos x\). This wave arises from plotting the cosine of different angles, producing a smooth, oscillating curve. The wave-like shape of the cosine function highlights its nature, continuously fluctuating between its peak values.
The fundamental characteristics of a cosine wave include its amplitude, which refers to the height of its peaks and troughs relative to the x-axis. For the basic function \(y = \cos x\):

• The amplitude is 1, because the maximum value is 1.
• The minimum value is -1.

This oscillation captures the essence of periodic behavior, smoothly transitioning through crucial turning points to form intersections with the x-axis.

Function Period

A function's period is the length of one complete cycle of its wave. For \(y = \cos x\), this period is \(2\pi\) radians. This indicates the point where the pattern begins to repeat itself.
The concept of a period is central to understanding any trigonometric function's repetition:

• If you visualize or sketch the cosine wave from 0 to \(2\pi\), you will see it completes a single cycle.
• After \(2\pi\), the cycle starts over, exactly replicating the previous one.

The ability to predict this pattern over any range of x-values makes periodic functions like the cosine function particularly useful in modeling alternating patterns and cycles found in numerous natural and scientific phenomena.

Key Points Plotting

Plotting key points is crucial in graphing the function \(y = \cos x\) accurately. These key points mark critical shifts in the cosine wave, indicating where it hits its maximum, minimum, and zero-crossings. Let's break these moments down:

• At \(x = 0\): \(y = 1\) (maximum)
• At \(x = \pi/2\): \(y = 0\) (crossing point)
• At \(x = \pi\): \(y = -1\) (minimum)
• At \(x = 3\pi/2\): \(y = 0\) (another crossing point)
• At \(x = 2\pi\): \(y = 1\) (back to maximum)

By plotting these points, you can visualize the shape of the curve. Once the key points are in place, connecting them with a smooth, continuous curve will accurately sketch the cosine wave's typical form. This approach simplifies the drawing process and ensures the wave's periodicity and symmetry are maintained.